223
1 INTRODUCTION
For safety of navigation, the ships are obliged to
comply with the International Regulations for
Preventing Collisions at Sea (COLREG). However,
these Rules refer only to two ships and under the
conditionsofgoodvisibility.
In the situation of a restricted visibility the
Regulations only specify recommendations of a
general nature and are not ab
le to consider all the
necessary conditions which determine the passing
course[1,2,3,4,6,7].
Consequently,theactualprocessofashippassing
other ships very often occurs in conditions of
uncertainty and conflict accompanied by an
inadequateco‐operationoftheshipswithinCOLREG
Rules.
It is, therefore, reasonab
le to investigate the
methodsofashipsafetyhandlingusingprinciplesof
the theory based on optimal control and differential
games[5,11,12,14,16,26,27].
2 CONTROLPROCESS
Theprocessofhandlingashipasamultidimensional
dynamicobjectdependsboth on theaccuracy of the
details concerning the current navigational situation
obtained from the Automa
tic Radar Plotting Aids
ARPA anti‐collision system and on the form of the
processmodelusedforthecontrolsynthesis[15,20].
The ARPA system ensures monitoring of atleast
20j encountered ships, determining their movement
parameters (speed V
j, course ψj) and elements of
approaching to own ship moving with speed V and
course ψ to satisfy
jj
DCPAD
min,
‐ Distance of
theClosestPointofApproach,and
jj
TCPA
T
min,
‐ Time to the Closest Point of Approa
ch and also
assesstheriskofcollisionr
j(Fig.1).
Analysis of Methods of Determining the Safe Ship
Trajectory
J
.Lisowski
GdyniaMaritimeUniversity,Gdynia,Poland
ABSTRACT: The paper describes six methods of optimal and game theory andartificial neural network for
synthesisofsafecontrolincollisionsituationsatsea.Theapplicationofoptimalandgamecontrolalgorithmsto
determinetheownshipsafetrajectoryduringthepassingofotherencounteredshipsingoodandrestricted
visibilit
yatseaispresented. Thecomparisonofthesafeshipcontrolincollisionsituation:multi‐stepmatrix
non‐cooperativeandcooperativegames,multi‐stagepositional non‐cooperative and cooperative games have
been introduced. The considerations have been illustrated with examples of computer simulation of the
algorit
hmstodeterminesafeofownshiptrajectoriesinanavigationalsituationduringpassingofeightmet
ships.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 10
Number 2
June 2016
DOI:10.12716/1001.10.02.05
224
Figure1.Thesituationofownshippassingeight
encounteredshipsatsea.
The model of the process consists both of the
kinematicsandthedynamicsoftheship’smovement,
the disturbances, the strategies of the encountered
ships and the quality control index of the own ship
[10,24].
The diversity of possible models directly affects
the synthesis of the ship’s control algorithms which
are
afterwards affected by the ship’s control device,
directlylinkedtotheARPAsystemandconsequently
determines the effects of safe and optimal control
[3,28].
3 COMPUTERPROGRAMSFORDETERMINING
THESAFEOWNSHIPTRAJECTORY
3.1 ComputerprogramSTATOPTofstaticoptimization
Goalcontrolfunctionhasform:
mjLtxLI
k
UUu
m
j
j
...,,2,1,)(
min
00
1
,000
(1)
L refers to the continuous function of the
manoeuvring goal of the own ship, describing the
distance of the ship at the initial moment t
0 to the
nearestturningpointT
ionthereferencerouteofthe
voyageandd
iisthefinaldeviationsafetrajectoryof
thereferencetrajectory(Fig.1).
3.2 ComputerprogramDYNOPTofdynamic
optimizationwithneuralstateconstraints
Determination of the optimal control of the ship in
termsofanadoptedindexofthecontrolqualitymay
be performed by applying Bellmanʹs principle of
optimization. The constraints for the state variables
andthecontrolvaluesgeneratetheneuralconstraints
procedureinthecomputerprogram[17,23].
The optimal time for the ship to go through k
stagesisasfollows:
1, 2 2, 2
1
,
min , 3,4,...,
kk
kkk
uu
tttkK
(2)
3.3 ComputerprogramMATGAM_Cofmulti‐step
cooperativematrixgame
ThematrixgameR
)],([
0
ssr
jj
includesthevalue
a collision risk r
j with regard to the determined
strategies s
0 of the own ship and those sj of the j‐th
encounteredship[8,13,18,19,21,22].
Thevalueoftheriskofthecollisionrjisdefinedas
thereferenceofthecurrentsituationoftheapproach
described by the parameters
min,j
D
and
min,j
T
, to
the assumed assessment of the situation defined as
safeanddeterminedbythesafedistanceofapproach
D
s and the safe time Ts – which are necessary to
execute a manoeuvring to avoide collision with
consideration actual distance D
j between own ship
andj‐thencounteredship[25,29].
Asaresultcontrolgoalfunctionhasform:
0
0
min min
j
j
ss
I
r
(3)
3.4 ComputerprogramMATGAM_NCofmulti‐step
non‐cooperativematrixgame
Goal function (3) for non‐cooperative matrix game
hastheform:
0
0
min max
j
j
s
s
I
r
(4)
3.5 ComputerprogramPOSGAM_Cofmulti‐stage
cooperativepositionalgame
The optimal control of the own ship
0
()ut
is
determinedfromthecondition:
0, 0,
00 0,
1
00
()
()
1,2,...,
min min min
m
jj j jj
j
j
k
uU u U u
uU U
I
Lx t L
jm
(5)
Thevalueofcontroliscalculatedateachdiscrete
stageoftheship’smovementbyapplyingtheSimplex
method to solve the problem of the triple linear
programming, assuming the relationship (5) as the
goalfunctionandthecontrolconstraints.
3.6 ComputerprogramPOSGAM_NCofmulti‐stage
non‐cooperative
positionalgame
Goalfunction(5)fornon‐cooperativepositionalgame
hastheform:
225
0, 0,
00 0,
1
00
()
()
1,2,...,
max
min min
m
jjj
jj
j
j
k
uUu
uU
uU U
I
Lx t L
jm
(6)
4 COMPUTERSIMULATION
Computersimulationofsixprogramswascarriedout
in Matlab/Simulink software on an example of the
real navigational situation of passing m=8
encountered ships in good visibility D
s=0,5 nm and
restrictedvisibilityD
s=2,0nm(nauticalmiles)(Figures
2‐13).
Figure2. Computer simulation of static optimization
STATOPT program determining of the safe own ship
trajectoryinsituationofpassingeightencounteredshipsin
goodvisibilityatsea,D
s=0,5nm,di=0,72nm(nauticalmile).
Figure4. Computer simulation of dynamic optimization
DYNOPT program determining of the safe own ship
trajectoryinsituationofpassingeightencounteredshipsin
goodvisibilityatsea,D
s=0,5nm,di=0nm(nauticalmile).
Figure3. Computer simulation ofSTATOPT program
determiningofthesafe own ship trajectory insituationof
passing eight encountered ships in restricted visibility at
sea,D
s=2,0nm,di=2,35nm(nauticalmile).
Figure5. Computer simulation of dynamic optimization
DYNOPT program determining of the safe own ship
trajectoryinsituationofpassingeightencounteredshipsin
restricted visibility at sea, D
s=2,0 nm, di=5,0 nm (nautical
mile).
226
Figure6. Computer simulation of multi‐step cooperative
matrixgameMATGAM_Cprogramdeterminingofthesafe
own ship trajectory in situation of passing eight
encounteredships ingoodvisibilityatsea,D
s=0,5nm,di=7,2
nm(nauticalmile).
Figure8. Computer simulation of multi‐step non‐
cooperative matrix game MATGAM_NC program
determiningofthesafe own ship trajectory insituationof
passing eight encountered ships in good visibility at sea,
D
s=0,5nm,di=8,8nm(nauticalmile).
Figure7. Computer simulation of multi‐step cooperative
matrixgameMATGAM_Cprogramdeterminingofthesafe
own ship trajectory in situation of passing eight
encounteredships inrestrictedvisibilityatsea,D
s=2,0nm,
d
i=7,8nm(nauticalmile).
Figure9. Computer simulation of multi‐step non‐
cooperative matrix game MATGAM_NC program
determiningofthesafe own ship trajectory insituationof
passing eight encountered ships in restricted visibility at
sea,D
s=2,0nm,di=9,2nm(nauticalmile).
227
Figure10. Computer simulation of multi‐stage cooperative
positional game POSGAM_C program determining of the
safe own ship trajectory in situation of passing eight
encountered ships in good visibility at sea, D
s=0,5 nm,
d
i=1,14nm(nauticalmile).
Figure12. Computer simulation of multi‐stage non‐
cooperative positional game POSGAM_NC program
determiningofthesafe own ship trajectory insituationof
passing eight encountered ships in good visibility at sea,
D
s=0,5nm,di=5,68nm(nauticalmile).
Figure11. Computer simulation of multi‐stage cooperative
positional game POSGAM_C program determining of the
safe own ship trajectory in situation of passing eight
encounteredships inrestrictedvisibilityatsea,D
s=2,0nm,
d
i=2,40nm(nauticalmile).
Figure13. Computer simulation of multi‐stage non‐
cooperative positional game POSGAM_NC program
determiningofthesafe own ship trajectory insituationof
passing eight encountered ships in restricted visibility at
sea,D
s=2,0nm,di=10,57nm(nauticalmile).
5 CONCLUSIONS
Thesynthesisofanoptimalandgamecontrolonthe
models of: static, dynamic with neural network,
multi‐step matrix game and multi‐stage positional
game makes it possible to determine the safe game
trajectory of the own ship in situations when she
passesagreaterjnumberof
theencounteredobjects.
The trajectory has been described as a certain
sequenceofmanoeuvreswiththecourseandspeed.
The computer programs designed in the Matlab
alsotakesintoconsiderationthefollowing:COLREG
228
Rules,advancetimeforamanoeuvrecalculatedwith
regard to the ship’s dynamic features and the
assessmentofthefinaldeviationbetweentherealand
referencetrajectories.
Theessentialinfluencetoformofsafeandoptimal
trajectory and value of deviation between real and
reference trajectories has a degree of
cooperation
betweenownandencounteredships.
The computer programs provides a formal
modeloftheactualdecision‐makingprocessleading
ship navigator and can be used with the system of
computer‐aided navigator when deciding to
manoeuverincaseofcollisionatsea.
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